6.849: Geometric Folding Algorithms: Linkages, Origami, Polyhedra (Fall 2012)

Prof. Erik Demaine       TA: Jayson Lynch


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[+] Folding motions: trouble with holes.
Linkages to sign your name: sliding joints, contraparallelogram bracing, higher dimensions, semi-algebraic sets, splines.
Geometric construction: straight edge and compass, origami axioms, angle trisection, cube doubling.

This class addresses the following questions:
  • Why paper with holes makes it hard to convert folded states to folding motions
  • How to simulate sliding joints with regular linkages
  • How the contraparallelogram gets braced (and why)
  • Kempe Universality Theorem in higher dimensions
  • Drawing semi-algebraic sets, e.g., splines, by taking intersections and unions of constructions

In addition, we cover the new topic of geometric construction via origami axioms, mentioned briefly at the end of Lecture 10. Single-fold origami can solve all cubic equations, unlike straight edge and compass which can solve (only) all quadratic equations. Two-fold origami can e.g. quintisect an angle, while n-fold origami can solve any polynomial.

We will also assemble our (nonexistent) square hypars from Class 9 and assemble them into a hyparhedron sculpture.

Download Video: 360p, 720p

Handwritten notes, page 1/4[previous page][next page][PDF]

Handwritten notes, page 1/4[previous page][next page][PDF]

Slides, page 1/21[previous page][next page][PDF]

Slides, page 1/21[previous page][next page][PDF]

The video above should play if your web browser supports either modern Flash or HTML5 video with H.264 or WebM codec. The handwritten notes and slides should advance automatically. If you have any trouble with playback, email Erik.